Probability of Drawing a Card from a Deck

Question

I like to play Magic: The Gathering, and I'm interested in calculating the probability of certain things in the game.

After drawing 7 cards from a 60 card deck, what is the probability that draw will contain at least one of x, where x is a card having y copies?

My best effort has been with the following formula:

$$ \binom{y}{1}\binom{60 - y}{6}/\binom{60}{7} $$

There are deck analyzers that will generate the answer for a simple 7-card draw (I'm wanting to get the formula though). Our answers are the same for y=1, but they start to deviate for y=2, y=3, etc. which has put some doubt in this formula.

What's the proper way to solve this?

Answer 1

The probability is $$ 1-\frac{60-y}{60}\frac{59-y}{59}\cdots\frac{54-y}{54} $$ since $$ \frac{60-y}{60}\frac{59-y}{59}\cdots\frac{54-y}{54} $$ is the probability that you don't draw one of the $y$ copies of $x$ on the first draw, second, $\dots$, or seventh draw.